Polynomial and rational functions

1. THE SENTENCE THE EXAM GRADES, AND THE VERBS THAT ASK FOR IT

2. CHANGE IN TANDEM: THE VOCABULARY OF A GRAPH, DEFINED BY RATES

A function maps each input to exactly one output; the inputs are the domain, the outputs the range; the input variable is independent, the output dependent. The image of an input is its one output; the preimage of an output is the set of inputs that produce it. Two functions are equal when they share a domain and agree at every input.

On an interval, $f$ is increasing if larger inputs always give larger outputs ($a<b$ implies $f\left(a\right)<f\left(b\right)$) and decreasing if larger inputs always give smaller outputs. The graph is concave up where the rate of change is increasing and concave down where the rate of change is decreasing — a definition about rates, not about height, and the one students reverse. The graph meets the $x$-axis where the output is zero; those inputs are the zeros.

The sentence for the first stretch, in the CED’s form: on the interval from $x=4$ to $x=38$, as $x$ increases, $y$ increases at an increasing rate; the graph is concave up. Every clause is a claim about rates, and every claim can be checked from a table.

3. RATES OF CHANGE: AVERAGE, INSTANTANEOUS, AND CHANGING

The average rate of change of $f$ over $[a,b]$ is the constant rate that would produce the same change in output over that interval:

the slope of the secant through $(a,f(a\left)\right)$ and $(b,f(b\left)\right)$, in output units per input unit. The rate of change at a point is how fast the output would change if the input changed there; it is approximated by average rates over small intervals containing the point, and two points’ rates are compared that way. A positive rate: the two quantities move together; negative: one rises as the other falls.

How the rate changes is the shape. Over consecutive equal-length intervals:

• a linear function has the same average rate on every interval — the rates change at a rate of zero;
• a quadratic function has average rates that form a linear pattern — the rates change at a constant rate, so the second differences are constant;
• when the rates over equal steps are increasing, the graph is concave up; decreasing, concave down.

The rates fall by 32 every step: a quadratic, concave down, and the $-32$ is $2\times (-16)$, twice the leading coefficient. That table is the whole of Topic 1.3.

4. POLYNOMIALS AND THEIR RATES OF CHANGE: EXTREMA AND INFLECTION

A polynomial of degree $n$ is $p\left(x\right)=a_n{x}^n+\cdots +a_{1}x+a_0$ with $a_n\ne 0$; $a_n{x}^n$ is the leading term. Where $p$ switches from increasing to decreasing it has a local maximum; from decreasing to increasing, a local minimum; a restricted domain’s included endpoint can be one too. A local extremum that beats every other output is global.

Three facts the exam tests as reasons:

• between any two distinct real zeros there is at least one local maximum or minimum — the graph has to turn to come back;
• a polynomial of even degree has a global maximum or a global minimum (both ends go the same way); a quadratic’s is its vertex;
• a point of inflection is where the rate of change switches from increasing to decreasing or back — where concavity changes.

5. ZEROS: FACTORS, MULTIPLICITY, COMPLEX PAIRS, $n$TH DIFFERENCES, EVEN AND ODD

$a$ is a zero of $p$ when $p\left(a\right)=0$; for a real $a$, $(x-a)$ is a factor of $p$ if and only if $a$ is a zero. A factor repeated $n$ times gives a zero of multiplicity $n$, and a polynomial of degree $n$ has exactly $n$ complex zeros counting multiplicity. A real zero is an $x$-intercept and an endpoint of the intervals that solve $p\left(x\right)\ge 0$ or $p\left(x\right)\le 0$. A non-real zero $a+\mathrm{bi}$ brings its conjugate $a-\mathrm{bi}$ with it.

Multiplicity is the shape at the zero. Odd multiplicity: the graph crosses. Even multiplicity: the outputs keep their sign on both sides, so the graph is tangent to the axis and bounces.

The degree from a table. Over equal-interval inputs, take successive differences of the outputs; the degree is the first $n$ for which the $n$th differences are constant. (Card 3’s quadratic table is constant at the second round.)

Even and odd functions. An even function satisfies $f(-x)=f\left(x\right)$ and is symmetric across the $y$-axis; $p\left(x\right)=a_n{x}^n$ with $n$ even is one. An odd function satisfies $f(-x)=-f\left(x\right)$ and is symmetric about the origin; $a_n{x}^n$ with $n$ odd is one. Most polynomials are neither; the test is the algebra, not the degree.

6. END BEHAVIOR OF A POLYNOMIAL: THE LEADING TERM WINS

As $x$ increases or decreases without bound, a nonconstant polynomial’s output increases or decreases without bound, and the notation is a limit: $\underset{x\to \infty }{\lim }p\left(x\right)=\infty$ or $-\infty$, and the same toward $-\infty$. Which way is decided by the degree and the sign of the leading term, because for large $\left|x\right|$ the leading term dominates every lower-degree term.

Even degree: both ends the same way, like $x^2$. Odd degree: opposite ways, like $x^3$. The sign of $a_n$ flips the picture. Nothing else in the polynomial has a vote at the ends.

7. END BEHAVIOR OF A RATIONAL FUNCTION: THE QUOTIENT OF LEADING TERMS

A rational function is a quotient of two polynomials, and for large $\left|x\right|$ each polynomial is dominated by its leading term, so the rational function is dominated by the quotient of the leading terms. Three cases:

• the numerator dominates (higher degree): the quotient of leading terms is a nonconstant polynomial, and $r$ has that polynomial’s end behavior; if it is linear, the graph has a slant asymptote parallel to that line (division gives the line itself);
• neither dominates (equal degrees): the quotient is a constant, the horizontal asymptote $y=$ ratio of leading coefficients;
• the denominator dominates: the quotient is a constant over a polynomial, and the horizontal asymptote is $y=0$.

When $y=b$ is a horizontal asymptote the outputs get and stay arbitrarily close to $b$ as $\left|x\right|$ grows: $\underset{x\to \infty }{\lim }r\left(x\right)=b$, $\underset{x\to -\infty }{\lim }r\left(x\right)=b$. The graph may cross a horizontal asymptote in the middle; the asymptote is a statement about the ends.

8. ZEROS, VERTICAL ASYMPTOTES AND HOLES: THE MULTIPLICITY CONTEST

Factor the numerator and the denominator. Then, at each real zero of either:

• a zero of the numerator only, inside the domain, is a zero of $r$ — an $x$-intercept;
• a zero of the denominator whose multiplicity there is greater than in the numerator is a vertical asymptote $x=a$: the denominator is arbitrarily close to $0$, so $r$ increases or decreases without bound, and each side gets its own limit — $\underset{x\to a^{+}}{\lim }r\left(x\right)=\pm \infty$, $\underset{x\to a^{-}}{\lim }r\left(x\right)=\pm \infty$;
• a zero of the denominator whose multiplicity in the numerator is at least as large is a hole at $x=c$: the factors cancel, and the hole sits at $(c,L)$ where $L$ is the value the simplified expression approaches — $\underset{x\to c}{\lim }r\left(x\right)=L$, the same from both sides.

The real zeros of both polynomials are the endpoints and asymptotes of the intervals that solve $r\left(x\right)\ge 0$ or $r\left(x\right)\le 0$: a sign chart with every one of them marked.

CHECK 2 — KEEP THE EXCLUDED INPUT

A hole, and the zero is at $x=-2$. The $(x-1)$ cancels, so for every $x\ne 1$ the function equals $x+2$ — but cancelling is something you did to the expression, not to the function, and $x=1$ was never in the domain to begin with. So the graph is the line $y=x+2$ with the point $\left(\mathrm{1,3}\right)$ lifted out of it. It is not a vertical asymptote, because nothing is left in the denominator to blow up; and it is not a zero, because a zero is a place the function takes the value $0$, and at $x=1$ the function takes no value at all. The zero is where the surviving numerator vanishes: $x+2=0$, so $x=-2$. The habit that protects all three answers is one line long — write the original denominator’s zeros down before you simplify anything, and carry that list to the end.

9. EQUIVALENT FORMS: WHAT EACH ONE SHOWS, DIVISION, AND THE BINOMIAL THEOREM

The factored form shows real zeros — and so $x$-intercepts, holes, vertical asymptotes, the domain and, for a rational function, much of the range. The standard form shows the leading term, and so end behavior. Different forms of one function answer different questions, and the exam asks you to pick.

Polynomial long division writes $f\left(x\right)=g\left(x\right)q\left(x\right)+r\left(x\right)$ with the remainder’s degree below the divisor’s — the same relationship as $17=5\cdot 3+2$. Its use in this unit is the slant asymptote: when a rational function’s numerator is one degree above its denominator, the quotient $q$ is a line and the graph approaches it, because $r\left(x\right)∕g\left(x\right)\to 0$ at the ends.

The binomial theorem expands ${(a+b)}^n$ with the $n$th row of Pascal’s triangle as coefficients — for $n=4$, the row $\mathrm{1,4,6,4,1}$:

so ${(x+c)}^n$ expands without multiplying it out $n$ times. Each term’s exponents sum to $n$, and the coefficients are symmetric.

10. TRANSFORMATIONS: FOUR MOVES, APPLIED IN ORDER, TRACKED ON ONE POINT

The two inside the function work backwards — $f(x+3)$ moves left, $f\left(2x\right)$ compresses — because the input has to be smaller, or arrive sooner, to produce the same output. The set of points before the transformation is the preimage, after it the image, and combined transformations apply in order: dilate, then translate, unless the parentheses say otherwise. The domain and range of the image can differ from the parent’s — $\frac{1}{x-3}+1$ has domain $x\ne 3$ and range $y\ne 1$, neither of which is $\frac{1}{x}$’s.

11. CHOOSING A MODEL, AND SAYING WHAT IT ASSUMES

The function type is read from the rates in the data or the scenario, and from its geometry:

Assumptions and restrictions are part of the model, and the exam asks for them: what the model assumes stays constant; how it assumes the quantities change together; a domain restriction from the mathematics or the context (a length cannot be negative; a sheet 20 cm wide allows corner squares under 10 cm); a range restriction such as rounding to whole units. A model with its restrictions unstated is incomplete.

12. BUILDING THE MODEL AND USING IT, WITH UNITS

Three ways to build one. From restrictions: zeros give factors, one more point fixes the leading coefficient — $p\left(t\right)=at(t-4)(t-10)$ with $p\left(2\right)=12$ gives $a=\frac{3}{8}$. From a parent: a transformation of $x^2$, $x^3$ or $\frac{1}{x}$ moved and scaled to the scenario. From data: a linear, quadratic, cubic or quartic regression on the calculator, chosen by the pattern of the rates, reported with its coefficients to three decimal places. A piecewise model combines techniques over intervals; a rational model fits inverse proportion, $I\left(d\right)=k∕d^2$, with $k$ fixed by one observation.

Applying it means answering the question that was asked: a predicted value, a rate of change, an average rate over an interval, a change in the rate — each with units extracted from the context (cubic centimeters, lumens, feet per second) and each read against the domain restriction. A prediction outside the domain is not a prediction the model makes.